2024 AMC 10A Problem 8

Problem 8 of 25EasierAlgebra

Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at

1{:}00

PM and were able to pack

4,

3,

and

3

packages, respectively, every

3

minutes. At some later time, Daria joined the group, and Daria was able to pack

5

packages every

4

minutes. Together, they finished packing

450

packages at exactly

2{:}45

PM. At what time did Daria join the group?

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Solution

From

1{:}00

2{:}45

105

minutes. Amy, Bomani, and Charlie pack

4 + 3 + 3 = 10

packages every

3

minutes, so

\tfrac{10}{3}

per minute, which is

\tfrac{10}{3} \cdot 105 = 350

packages. That leaves

450 - 350 = 100

for Daria, who packs

\tfrac54

per minute and so needs

\frac{100}{\tfrac54} = 80

minutes. She worked the last

80

minutes, joining

105 - 80 = 25

minutes after

1{:}00.

That’s

1{:}25

PM. Therefore, the answer is A.